Criteria Of Existence Of Global Solutions For Nonlinear Wave Equations With Damping

This thesis studies two nonlinear wave equations: nonlinear beam equations with and without a damping,and fractional wave equations with a damping.By using the contracting mapping principle,Gronwall lemma and various interpolation estimates,the wellposedness and the sharp criteria for the existence of global solutions to the above two nonlinear wave equations are studied.This thesis is divided into the following four chapters:In Chapter 1,the backgrounds,contents and main results of this thesis are summarized.In Chapter 2,the working spaces and Gagliardo-Nirenberg inequality are introduced.According to the structure and characteristics of the two types of nonlinear wave equations,the corresponding energy conservation laws are constructed.In Chapter 3,the initial value problem of nonlinear beam equations with and without a damping is discussed.First,the existence and uniqueness of the local solution to the nonlinear beam equation with a damping are proved by the contracting mapping principle.Then,sufficient condition for the existence of global solutions is obtained by Gronwall lemma and the continuity principle.Finally,the Gagliardo-Nirenberg inequality and the variational characteristics of the fourth-order nonlinear elliptic equation are applied to construct the invariant sets of the nonlinear beam equation with and without a damping,respectively.Moreover,the sharp criteria for the existence of global solutions of the corresponding nonlinear beam equation are obtained.In Chapter 4,the initial value problem of the fractional wave equation with a damping is studied.By using the Cauchy-Schwartz inequality,the sufficient condition for the existence of blow up solutions is obtained.Then,the Gagliardo-Nirenberg inequality and the variational characteristics of the fractional nonlinear elliptic equation are applied to construct the invariant sets of the equation,and then the sharp criteria for the existence of global solutions of the fractional degenerate Kirchhoff-type wave equations with a damping is obtained.